2005/07/31 by Paweł J. Szabłowski, P. J. Szabł owski
Mathematics · #Combinatorics #Context (archaeology) #Discrete mathematics #Generalization #Hermite polynomials #Markov Chains and Monte Carlo Methods #Mathematical analysis #Mathematical functions and polynomials #Mathematics #Ornstein–Uhlenbeck process #Pure mathematics #Random Matrices and Applications #Simple (philosophy) #Stochastic process #Wiener process #math.PR #msc:05A30 #msc:60G44 #msc:60J25 #msc:60K40
paper · pdf · doi:10.1137/s0040585x97985674
published as THEORY PROBAB. APPL. vol. 56(2012) nr. 4 pp. 634--659 · 25 pages
arxiv created 2011/10/21 · openalex publication_date 2012/01/01 · arxiv updated 2013/06/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We collect and prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein--Uhlenbeck (OU) processes. Although processes considered in this paper were defined either in a noncommutative probability context or through quadratic harnesses we define them once more as a “continuous time” generalization of a simple, symmetric, discrete time process satisfying simple conditions imposed on the form of its first two conditional moments. The finite dimensional distributions of the first one (say \bf X=( Xt)t≥ 0 called q-Wiener) depend on one parameter q∈(-1,1], and those of the second one (say \bf Y=(Yt)_t∈\bf R called (α,q)-Ornstein--Uhlenbeck) on two parameters (α,q)∈(0,∞)×(-1,1]. The first one resembles the Wiener process in the sense that for q=1 it is a Wiener process but also that for \vert q\vert <1 and ∀ n≥1: tn/2Hn( Xt/√(t) | q), where (Hn)n≥0 are the so-called q-Hermite polynomials, are martingales. However, it neither has independent increments not allows continuous sample path modification. The second one resembles the OU process. For q=1 it is a classical OU process. For \vert q\vert <1 it is also stationary with correlation function equal to exp(-α|t-s|) and has many properties resembling those of its classical version. We think that these processes are fascinating objects to study posing many interesting, open questions.